0
$\begingroup$

In Kollár and Mori's Birational Geometry of Algebraic Varieties, the authors say a Cartier divisor is big iff its birational pullback is big (Definition 2.59 below). But I can't understand. Maybe the varieties are proper by definition of big divisor.

And I know when the varieties are integral normal and proper, this can be done by Zariski main theorem and projectiive formula. Are conditions 'integral' and 'normal' necessary?

Thank you for any answer or comments.

$\endgroup$

2 Answers 2

3
$\begingroup$

In this book Kollar and Mori define big divisors only for proper (irreducible) varieties, so when they say that for a birational morphism $f:X\to Y$, the pullback $f^*D$ of a divisor $D$ on $Y$ is big if and only if $D$ is big, they probably mean $X$ and $Y$ are proper.

To see that $D$ is big if and only if $f^*D$ is, you can use Lemma 2.60. This says in particular that a Cartier divisor $E$ on a variety $Z$ is big if, and only if, the rational map $\phi_{kE}$ defined by some multiple of it has image of dimension equal to $\dim Z$. The point is that $X$ and $Y$ have isomorphic dense open subsets and the maps $\phi_{kD}$ and $\phi_{kf^*D}$ coincide on them.

$\endgroup$
1
$\begingroup$

I guess volume can also be defined on a compact kahler manifold according to Lazarsfelds book 2.2.53. On the other hand, if the manifold is not compact, then it might not be possible to always get an actual number for the volume (although I don't know any examples off the top of my head) (and D is big iff volume > 0, so you want the volume to be well-defined first).

$\endgroup$
3
  • 2
    $\begingroup$ How is this answer related to the question? $\endgroup$ Mar 11, 2015 at 23:15
  • $\begingroup$ It seems like he is asking why is "properness of the variety" included in the definition for big divisor... but recall that properness in algebraic variety sense is compactness in the complex analytic sense. Thus my answer was an attempt to describe why properness might be desirable in a definition for bigness. $\endgroup$
    – shen
    Mar 11, 2015 at 23:48
  • $\begingroup$ In addition, if the asker is in fact wondering about the normal hypothesis, I think volume is birationally invariant, c.f. 2.2.43 from the same book I mentioned. So using the definition of big iff vol > 0 in that case should help with that question as well. $\endgroup$
    – shen
    Mar 11, 2015 at 23:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.